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Section 1.8 Introduction Bootcamp

Subsection 1.8.1 Units

Subsection 1.8.2 Uncertainty, Significant Figures, Rounding

Subsection 1.8.3 Propagation of Uncertainty

Subsection 1.8.4 Order of Magnitude

Subsection 1.8.5 Dimensional Analysis

Subsection 1.8.6 Miscellaneous

In Astronomy, a unit of distance, called parsec (pc), is in common use. It is defined to be the distance at which an object of size 1 Astronomical Units (AU) \((\approx 150\times10^6\ \text{km})\) will subtend an angle of 1 arc-second, which is equal to \(1/3600\) of one degree. Two stars in a binary star system separated by \(8.6 \times 10^{14}\ \text{m}\) are seen to subtend an angle of \(0.2\ \text{arcsec}\text{.}\) How far away are the stars (a) in pc, and (b) in meters.

Hint

Answer

(a) \(2.9 \times 10^4\text{ pc}\text{;}\) (b) \(8.9 \times 10^{20}\text{ m}\text{.}\)

Solution

A \(5\text{-m}\) pole is attached to a boat. When the boat is \(4\text{ km}\) away from the shore, you can see only \(3\text{ m}\) of the pole above water. Use this information and assumption that Earth is a spherical ball to estimate the radius of earth.

Figure 1.8.32.
Hint

Answer

Solution

The British physicist G. I. Taylor argued that the radius \(R\) of a spherically symmetric nuclear explosion must depend on the energy \(E\text{,}\) the initial density of air \(D\text{,}\) and time \(t\) since explosion. Using dimensional analysis, find a formula for the radius at time t after the explosion. [Challenging problem]

Hint

Answer

\(R=CE^{1/5}D^{-1/5}t^{2/5}\text{,}\) where C is a dimensionless constant.

Solution

By dimensional analysis, find a formula for the oscillation frequency of a star of radius \(R\) and density \(D\text{.}\) Note that you will also need the dimensions of Newton's gravitational constant also, which is \([G_N]=[L]^3[T]^{-2}[M]^{-1}\text{.}\) [Challenging problem]

Hint

Answer

Solution